Are neural networks the only way to reach "true" artificial intelligence?
If by true AI, you mean 'like human beings', the answer is - no-one knows what the appropriate computational mechanisms (neural or otherwise) are or indeed whether we are capable of constructing them.

What Artificial Neural Nets (ANNs) do is essentially 'nonlinear regression' - perhaps this is not a sufficiently strong model to express humanlike behaviour. 

Despite the 'Universal function approximation' property of ANNs, what if human intelligence depends on some as-yet-unguessed mechanism of the physical world?

With respect to your question about "the only way":
Even if (physical) neural mechanisms somehow actually were the only route to intelligence (e.g. via Penrose's quantum microtubules), how could that be proved? 

Even in the formal world of mathematics, there's a saying that "Proofs of non-existence are hard". It scarcely seems conceivable that, in the physical world, it would be possible to demonstrate that intelligence could not arise by any other mechanism.

Moving back to computational systems, note that Stephen Wolfram made the interesting observation in his book 'A New Kind of Science' that many of the apparently distinct mechanisms he observed seem to be capable of 'Universal Computation', so in that sense there's nothing very particular about ANNs.